Maximal and minimal elements

In mathematics, especially in order theory, a maximal element of a subsetS of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S.

The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. While a partially ordered set can have at most one each maximum and minimum it may have multiple maximal and minimal elements.[1][2] For totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.

As an example, in the collection

S = {{d, o}, {d, o, g}, {g, o, a, d}, {o, a, f}}

ordered by containment, the element {d, o} is minimal as it contains no sets in the collection, the element {g, o, a, d} is maximal as there are no sets in the collection which contain it, the element {d, o, g} is neither, and the element {o, a, f} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for S.

In general ≤ is only a partial order on S. If m is a maximal element and s∈S, it remains the possibility that neither s≤m nor m≤s. This leaves open the possibility that there are many maximal elements.

Example 3: In the fencea1 < b1 > a2 < b2 > a3 < b3 > ..., all the ai are minimal, and all the bi are maximal, see picture.

Example 4: Let A be a set with at least two elements and let S={{a}: a∈A} be the subset of the power setP(A) consisting of singletons, partially ordered by ⊂. This is the discrete poset—no two elements are comparable—and thus every element {a}∈S is maximal (and minimal) and for any distinct a′,a″ neither {a′} ⊂ {a″} nor {a″} ⊂ {a′}.

Power set of {x,y,z}, partially ordered by ⊂. Its greatest element, {x,y,z}, is its only maximal element.

It looks like m{\displaystyle m} should be a greatest element or maximum but in fact it is not necessarily the case: the definition of maximal element is somewhat weaker. Suppose we find s∈S{\displaystyle s\in S} with maxS≤s{\displaystyle \max S\leq s}, then, by the definition of greatest element, s≤maxS{\displaystyle s\leq \max S} so that s=maxS{\displaystyle s=\max S}. In other words, a maximum, if it exists, is the (unique) maximal element.

The converse is not true: there can be maximal elements despite there being no maximum. Example 3 is an instance of existence of many maximal elements and no maximum. The reason is, again, that in general ≤{\displaystyle \leq } is only a partial order on S{\displaystyle S}. If m{\displaystyle m} is a maximal element and s∈S{\displaystyle s\in S}, it remains the possibility that neither s≤m{\displaystyle s\leq m} nor m≤s{\displaystyle m\leq s}.

If there are many maximal elements, they are in some contexts called a frontier, as in the Pareto frontier.

Of course, when the restriction of ≤{\displaystyle \leq } to S{\displaystyle S} is a total order, the notions of maximal element and greatest element coincide. Let m∈S{\displaystyle m\in S} be a maximal element, for any s∈S{\displaystyle s\in S} either s≤m{\displaystyle s\leq m} or m≤s{\displaystyle m\leq s}. In the second case the definition of maximal element requires m=s{\displaystyle m=s} so we conclude that s≤m{\displaystyle s\leq m}. In other words, m{\displaystyle m} is a greatest element.

Finally, let us remark that S{\displaystyle S} being totally ordered is sufficient to ensure that a maximal element is a greatest element, but it is not necessary.
For example, every power set P(S) of a set S has only one maximal element, viz. S itself, which is also the unique greatest element; but almost no power set is totally ordered, cf. picture.

In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation applies not only to totally ordered subsets of any poset, but also to their order theoretic generalization via directed sets. In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. Any maximal element of such a subset will be unique (unlike in a poset). Furthermore, this unique maximal element will also be the greatest element.

Similar conclusions are true for minimal elements.

Further introductory information is found in the article on order theory.

In economics, one may relax the axiom of antisymmetry, using preorders (generally total preorders) instead of partial orders; the notion analogous to maximal element is very similar, but different terminology is used, as detailed below.

In consumer theory the consumption space is some set X{\displaystyle X}, usually the positive orthant of some vector space so that each x∈X{\displaystyle x\in X} represents a quantity of consumption specified for each existing commodity in the
economy. Preferences of a consumer are usually represented by a total preorder⪯{\displaystyle \preceq } so that x,y∈X{\displaystyle x,y\in X} and x⪯y{\displaystyle x\preceq y} reads: x{\displaystyle x} is at most as preferred as y{\displaystyle y}. When x⪯y{\displaystyle x\preceq y} and y⪯x{\displaystyle y\preceq x} it is interpreted that the consumer is indifferent between x{\displaystyle x} and y{\displaystyle y} but is no reason to conclude that x=y{\displaystyle x=y}, preference relations are never assumed to be antisymmetric. In this context, for any B⊂X{\displaystyle B\subset X}, we call x∈B{\displaystyle x\in B} a maximal element if

y∈B{\displaystyle y\in B} implies y⪯x{\displaystyle y\preceq x}

and it is interpreted as a consumption bundle that is not dominated by any other bundle in the sense that x≺y{\displaystyle x\prec y}, that is x⪯y{\displaystyle x\preceq y} and not y⪯x{\displaystyle y\preceq x}.

It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set. However, when ⪯{\displaystyle \preceq } is only a preorder, an element x{\displaystyle x} with the property above behaves very much like a maximal element in an ordering. For instance, a maximal element x∈B{\displaystyle x\in B} is not unique for y⪯x{\displaystyle y\preceq x} does not preclude the possibility that x⪯y{\displaystyle x\preceq y} (while y⪯x{\displaystyle y\preceq x} and x⪯y{\displaystyle x\preceq y} do not imply x=y{\displaystyle x=y} but simply indifference x∼y{\displaystyle x\sim y}). The notion of greatest element for a preference preorder would be that of most preferred choice. That is, some x∈B{\displaystyle x\in B} with

y∈B{\displaystyle y\in B} implies y≺x.{\displaystyle y\prec x.}

An obvious application is to the definition of demand correspondence. Let P{\displaystyle P} be the class of functionals on X{\displaystyle X}. An element p∈P{\displaystyle p\in P} is called a price functional or price system and maps every consumption bundle x∈X{\displaystyle x\in X} into its market value p(x)∈R+{\displaystyle p(x)\in {\mathbb {R} }_{+}}. The budget correspondence is a correspondence Γ:P×R+→X{\displaystyle \Gamma \colon P\times {\mathbb {R} }_{+}\rightarrow X} mapping any price system and any level of income into a subset

It is called demand correspondence because the theory predicts that for p{\displaystyle p} and m{\displaystyle m} given, the rational choice of a consumer x∗{\displaystyle x^{*}} will be some element x∗∈D(p,m){\displaystyle x^{*}\in D(p,m)}.

A subset Q{\displaystyle Q} of a partially ordered set P{\displaystyle P} is said to be cofinal if for every x∈P{\displaystyle x\in P} there exists some y∈Q{\displaystyle y\in Q} such that x≤y{\displaystyle x\leq y}. Every cofinal subset of a partially ordered set with maximal elements must contain all maximal elements.

A subset L{\displaystyle L} of a partially ordered set P{\displaystyle P} is said to be a lower set of P{\displaystyle P} if it is downward closed: if y∈L{\displaystyle y\in L} and x≤y{\displaystyle x\leq y} then x∈L{\displaystyle x\in L}. Every lower set L{\displaystyle L} of a finite ordered set P{\displaystyle P} is equal to the smallest lower set containing all maximal elements of L{\displaystyle L}.